Electrostatic increase of particle mass

[Andrew Mason]

Then radiation by a single charge is either impossible or carries no energy?

This is a very good question. I would begin my answer with another question: does a charge ever radiate by itself? A similar related question would be: can a charge accelerate by itself?

I think the answer to both of my questions is that it cannot. It can accelerate only by interacting with another particle. Whether the acceleration is the result of the interaction that causes radiation or the cause of the radiation is another good question. Remarkably, this is still an unresolved question in science.

So in answer to your question, I would say that a single electron cannot produce radiation by itself. An electron can produce radiation only by interacting with other particles (eg. changing energy levels in an atom), in which case the energy comes from this interaction.

As a follow-up: if an electron could radiate by itself using the energy contained in its field, would the strength of the field (ie. the magnitude of the charge) not have to decrease?

AM

 

[Andrew Mason]

In the gravitational case the potential energy goes into/comes from the gravitational field. The net gravity of the two bodies is altered as well by the change.

I am not aware of any experimental evidence that the potential energy comes from or goes into the gravitational field. Are you?

In relation to the original post the mass of an electron does not change at all in any electric field. The field itself acquires an inertia (and gravitates as well).

That is not the question. The question is whether there is energy, hence inertia, in the field of a single electron.

If an electron is placed in an electric field, it has energy by virtue of its position. That means the mass of the electron and the charge/charges that create the field together are greater than their masses individually. But that does not mean that there is energy in the e-field of a single electron.

A charged capacitor is heavier and the center of mass of the added mass is in the space between the plates, not in the charges on either plate.

I am not aware of any experimental evidence that establishes where the added mass is physically located. Can you point me to such evidence?

AM

 

[Drakkith]

I think this is a result of a misunderstanding of what energy is.

 

[kof9595995]

This is incorrect. I would have replied to your earlier remark on this if I'd had any idea you were even remotely serious. The energy density in an E field is always (1/8π) E·E, and is just as real regardless of whether it comes from one charge or two charges. Energy is a local quantity, and yes it is meaningful to say where it is located. .

General Relativity has the electromagnetic field as a source of gravity including terms proportional to E^2, so the contribution of the (static) electric field to mass-energy is obvious here.

The field energy in an electrostatic field is well-defined so why would it not be so for the case of a single electron? The source of the field is irrelevant. The energy density is 1/2 epsilon E^2.

That is the energy density of a charge distribution. If you apply the same analysis to the field of an electron, you get a different result - the energy is infinite. This problem was studied by Feynman who concluded that the concept of the field was an unnecessary abstraction. It is a little difficult to see how an unnecessary abstraction it can be a source of inertia, which is quite real. See http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html"

AM

I think these are quite relevant to an old question I asked :https://www.physicsforums.com/showthread.php?t=397520
Now I tend to think in the framework of EM itself it is indeed impossible to test whether energy is localized by field, but perhaps general relativity provides us a way to test it?

 

[Antiphon]

If you assume that field energy is localized, it makes sense to assume field energy is localized with some functional relation to the field intensity. There's no experimental proof for this but GR says it should be so.

I went over the derivation of electrostatic potential energy E=1/2 epsilon E^2 from first principles beginning with two, then three then N charges.

You get a nested summation where indeed you don't count any of the self-terms. This means the field of a single electron is not part of the field energy calculation.

One is of course left wondering now what is to be concluded about the validity of taking the summation over to an integral of continuous charge density. It would seem there is a defect in the assumption of charges being point objects. These infinite self energies are rendered finite if we assume the electron is not a point particle. But I'm afraid we're lost in the woods beyond here. Or at least I am.

 

[Per Oni]

If an electron is placed in an electric field, it has energy by virtue of its position. That means the mass of the electron and the charge/charges that create the field together are greater than their masses individually. But that does not mean that there is energy in the e-field of a single electron. AM

In my opinion there is energy in the E-field of a single electron.
Consider next experiment: a neutral metal sphere is placed a good distance apart from an electron, however not so far as that the electron cannot “detect” this sphere. What will happen is that the electron accelerates towards and collides with this sphere and in doing so will warm up the sphere a tiny bit. This thermal energy is due to the disappearance of a part of the electronic field. This follows from the fact this electron can now be thought to be located in the centre of the sphere but its field can only be found outside the sphere. Therefore some of the electric field energy of this electron is converted to kinetic energy.

 

[Antiphon]

The problem is that the conducting surface introduces additional charges.

I'm with you, I think there is energy here but I'm unable to calculate how much.

Clasically its equal to the amount of energy it takes to "assemble" the charge of a single electron. I get that this energy is infinite if the electron has no size.

 

[Drakkith]

I literally don't understand how you could have energy in a single field without anything else to act upon. Doesn't that kind of bypass the definition of energy?

As for "assembling" the electron, isn't the energy required equal to it's rest mass energy equivalence? Hence the creation of an electron requires at least as much energy as its rest mass in particle collisions or in pair productions? (Or twice as much since you would create a positron as well)

 

[Per Oni]

The problem is that the conducting surface introduces additional charges.

That’s true but don’t forget that it’s this single electron which provides the energy required to separate these charges. Long before the electron actually hits the sphere energy will have been supplied by the electron. The sphere doesn’t supply any energy.

Clasically its equal to the amount of energy it takes to "assemble" the charge of a single electron. I get that this energy is infinite if the electron has no size.

That’s why once upon a time an electron had a classical radius. For various good reasons this has been frowned upon.

I literally don't understand how you could have energy in a single field without anything else to act upon. Doesn't that kind of bypass the definition of energy?

It doesn’t bypass this definition: Energy density = ½ E^2 epsilon zero

 

[Drakkith]

It doesn’t bypass this definition: Energy density = ½ E^2 epsilon zero

Got a link for more info on that?

 

[Antiphon]

I literally don't understand how you could have energy in a single field without anything else to act upon. Doesn't that kind of bypass the definition of energy?

As for "assembling" the electron, isn't the energy required equal to it's rest mass energy equivalence? Hence the creation of an electron requires at least as much energy as its rest mass in particle collisions or in pair productions? (Or twice as much since you would create a positron as well)

No, charge and mass in this case are not related. The energy needed to pull any amount of charge, in this case q, into a point is unbounded.

Like I said, if you lower an electron and positron together slowly on little strings, you will be able to extract some energy as the net field heads toward zero. Classically, you could extract it all as the point charges coincided and there would be no annihilation. There would be infinite energy available from this unless you assume non-zero size for the electron.

The rest mass is there but constitutes less energy than a free electron has vis-a-vis the experiment above. A free (widely separated) electron/positron pair plus the EM field together contain more energy than 2m0c^2.

As for E=1/2 epsilon E^2, that equation can't be applied to a single electron.

 

[Drakkith]

No, charge and mass in this case are not related. The energy needed to pull any amount of charge, in this case q, into a point is unbounded.

Like I said, if you lower an electron and positron together slowly on little strings, you will be able to extract some energy as the net field heads toward zero. Classically, you could extract it all as the point charges coincided and there would be no annihilation. There would be infinite energy available from this unless you assume non-zero size for the electron.

I don't see how this explains an isolated particle and its field.

The rest mass is there but constitutes less energy than a free electron has vis-a-vis the experiment above. A free (widely separated) electron/positron pair plus the EM field together contain more energy than 2m0c^2.

As for E=1/2 epsilon E^2, that equation can't be applied to a single electron.

Sure, if we look at it from the reverse situation, the creation of the pair would require more energy to get them to a distance. However I'm not sure the term energy is applicable with a single particle, as there is nothing for it to act upon.

 

[harrylin]

I literally don't understand how you could have energy in a single field without anything else to act upon. Doesn't that kind of bypass the definition of energy?

As for "assembling" the electron, isn't the energy required equal to it's rest mass energy equivalence? Hence the creation of an electron requires at least as much energy as its rest mass in particle collisions or in pair productions? (Or twice as much since you would create a positron as well)

Yes indeed, I already answered that in post #6 (with reference) but perhaps I did not use enough words.

Classically the rest energy of an electron is found to be approximately equal to the energy of its electric field. This is calculated from the energy that is required to add infinitely small amounts of charge to the electron, starting from zero charge.

Note: I also remember reading something about the energy that may be thought to be needed to keep an electron together and other such discussions one century ago, can anyone clarify this?

Elaborations:
- if we take the mass-energy equation to belong to classical physics (and at least in approximation this is the case), then the energy required to build up an electron's electric field (if that is the only thing in an electron that stores energy) should correspond to its rest mass.
- the exact relationship also depends on our assumption of the electron's charge density distribution. The concept of a literal point charge fails here, the obvious first approximation is a homogeneously charged sphere.

As all that gives reasonable numbers, one may count this as a success of the classical approach.

 

[Per Oni]

Got a link for more info on that?

From wiki:

Energy in the electric field
Main article: Electric energy
The electric field stores energy. The energy density of the electric field is given by

where ε is the permittivity of the medium in which the field exists, and E is the electric field vector.

No doubt you have seen this equation before but what puzzles me is why some people here think there’s an exception to this formula in case the field of a single electron is considered.

(sorry the formula didn't past but it's the same I've used)

 

[Drakkith]

Classically the rest energy of an electron is found to be approximately equal to the energy of its electric field. This is calculated from the energy that is required to add infinitely small amounts of charge to the electron, starting from zero charge.

What do you mean by "adding infinitely small amounts of charge"? How can you add charge or even calculate adding charge to an electron?

 

[Antiphon]

From wiki:


No doubt you have seen this equation before but what puzzles me is why some people here think there’s an exception to this formula in case the field of a single electron is considered.

(sorry the formula didn't past but it's the same I've used)

The equation is derived from assembling 2, 3, ...N charges. The equation is a double summation over i,j where you leave out the i=j terms. That's the key step.

After collecting terms and recognizing that the forces are symmetrical between charges, you can reduce the double sum to a single summation index. But by then all the self interaction terms have been excluded. The original double summation was undefined for a single charge because it excluded the i=j=1 interaction.